1. In a large, heavy-bottomed pot over medium-high heat, heat brown sugar, water, and salt. Cook about 10 to 12 minutes, until a smoking, dark brown caramel with a nutty aroma forms.

2. Immediately whisk the cream and milk into the caramel to stop the cooking. The mixture will steam and the caramel will seize, but will become smooth again as you whisk.

3. Bring mixture to a boil, then reduce heat to medium.

4. In a medium bowl, whisk together egg, egg yolks and cornstarch. Temper the hot caramel cream into the egg mixture by adding a cup of caramel at a time, whisking constantly. When half the caramel is incorporated, pour the egg mixture back into the remaining caramel and boil, whisking constantly, for about 2 minutes, until the custard is very thick.

5. Remove from heat and whisk in butter and rum. Strain custard through a fine mesh strainer to remove any lumps, place into a container with a lid and chill in the refrigerator for several hours or up to 3 days.